Blessing: God's favor and protection (Oxford Standard Dictionary)
It is He who confers blessing upon you, and His angels [ask Him to do so] that He may bring you out from darknesses into the light. And ever is He, to the believers, Merciful. (Qur'an 33:43)
The word Allah uses to describe his blessings upon us is 'yusalleeAAalaykum'=blessing upon you
Indeed, Allah confers blessing upon the Prophet, and His angels [ask Him to do so]. O you who have believed, ask [ Allah to confer] blessing upon him and ask [ Allah to grant him] peace. (Qur'an 33:56)
The word Allah uses to describe his blessings upon the Prophet (peace be upon him) is 'yusalloonaAAala'=blessing upon
So, it is Allah who blesses us. BUT,
Blessed is He in whose hand is dominion, and He is over all things competent (Qur'an 67:1)
The word Allah uses to describe himself is 'Tabaraka'= Blessed is He
So, is he blessing himself? Let's see further.
bless: to ask for God's help and protection for someone or something, or to call or make someone or something Holy (Cambridge Dictionary)
So, he is calling himself Holy here. In Arabic, some words don't have a very specific meaning but they change according to the context.
And We have sent down blessed rain from the sky and made grow thereby gardens and grain from the harvest (Qur'an 50:9)
The word used here to describe rain is 'mubarakan'=blessed (as we say Eid Mubarak=Blessed Eid)
Verily, man (disbeliever) was created very impatient; 19 ...Except those devoted to Salat (prayers) 22 (Qur'an 70:19 & 22)
The word used here is 'almusalleen'=devoted to Salat(who pray)
Not only these words, but other words e.g. 'Haq'=Truth or 'Haq'=Right(as in 'I don't have the right to speak') depends on the context and hence should not be confused.
SO, when we send salah to Allah, we ask him to bless us(have mercy us), when he sends salah, he is blessing us(showing mercy on us) and like wise on Prophet Muhammad (peace be upon him)...
MAY THE CREATOR GUIDE US ALL.